Multiple nontrivial solutions to a $p$-Kirchhoff equation

Anran  Li; Jiabao Su

doi:10.3934/cpaa.2016.15.91

Article Contents

2016, Volume 15, Issue 1: 91-102. Doi: 10.3934/cpaa.2016.15.91

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Multiple nontrivial solutions to a $p$-Kirchhoff equation

Anran Li^1, and
Jiabao Su^2,

1.
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi
2.
School of Mathematical Sciences, Capital Normal University, Beijing 100048

Received: December 31, 2014

Revised: March 31, 2015

Published: December 2015

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Abstract

In this paper, by computing the relevant critical groups, we obtain nontrivial solutions via Morse theory to the nonlocal $p$-Kirchhoff-type quasilinear elliptic equation \begin{eqnarray} (P)\quad\quad &&\displaystyle\bigg[M\bigg(\int_\Omega|\nabla u|^p dx\bigg)\bigg]^{p-1}(-\Delta_pu) = f(x,u), \quad x\in\Omega,\\ && u=0, \quad x\in \partial \Omega, \end{eqnarray} where $\Omega \subset \mathbb R^N$ is a bounded open domain with smooth boundary $\partial \Omega$ and $N \geq 3$.

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Mathematics Subject Classification: 35B38, 35D05, 35J50.

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